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A119011 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k valleys strictly above the x-axis (0<=k<=n-2; n>=2). A hill in a Dyck path is a peak at level 1.

%I A119011 #7 Apr 06 2019 09:29:42
%S A119011 1,1,1,2,3,1,3,8,6,1,5,18,23,10,1,8,38,70,54,15,1,13,76,186,215,110,
%T A119011 21,1,21,147,451,710,560,202,28,1,34,277,1025,2065,2269,1288,343,36,1,
%U A119011 55,512,2220,5480,7854,6321,2688,548,45,1,89,932,4634,13574,24227,25830
%N A119011 Triangle read by rows: T(n,k) is the number of hill-free Dyck paths of semilength n and having k valleys strictly above the x-axis (0<=k<=n-2; n>=2). A hill in a Dyck path is a peak at level 1.
%C A119011 Row sums yield the Fine numbers (A000957). T(n,0)=A000045(n-1) (the Fibonacci numbers). T(n,1)=A006478(n). Sum(k*T(n,k),k=0..n-2)=A119012(n)
%H A119011 E. Deutsch and L. Shapiro, <a href="https://doi.org/10.1016/S0012-365X(01)00121-2">A survey of the Fine numbers</a>, Discrete Math., 241 (2001), 241-265.
%F A119011 G.f.: G(t,z)=1/[1-zr(t,z)]-1, where r=r(t,z) is the Narayana function, defined by (1+r)(1+tr)z=r, r(t,0)=0. See Maple program for the explicit form of G(t,z).
%e A119011 T(5,2)=6 because we have uud|ud|uuddd, uuudd|ud|udd, uud|uudd|udd, uuud|ud|uddd, uuud|udd|udd and uud|uud|uddd (the valleys above the x-axis are marked with |).
%e A119011 Triangle starts:
%e A119011 1;
%e A119011 1,1;
%e A119011 2,3,1;
%e A119011 3,8,6,1;
%e A119011 5,18,23,10,1;
%p A119011 G:=2*t/(2*t+z*t+z-1+sqrt(z^2*t^2-2*z^2*t-2*z*t+z^2-2*z+1))-1: Gser:=simplify(series(G,z=0,15)): for n from 2 to 12 do P[n]:=sort(coeff(Gser,z^n)) od: for n from 2 to 12 do seq(coeff(P[n],t,j),j=0..n-2) od; # yields sequence in triangular form
%Y A119011 Cf. A000957, A000045, A006478, A119012.
%K A119011 nonn,tabl
%O A119011 2,4
%A A119011 _Emeric Deutsch_, May 08 2006